Linear Algebra: Course Overview and Key Concepts

Linear Algebra: Course Overview

Introduction

This study guide provides an overview of the main topics and learning objectives for a college-level Linear Algebra course (MATH 3307). Linear Algebra is a foundational subject in mathematics, focusing on vector spaces, linear transformations, matrices, and their applications. The course is designed to develop students' understanding of abstract mathematical structures and their practical uses in science, engineering, and data analysis.

Course Learning Objectives

Key Concepts and Skills

• Linear Transformation: A function between vector spaces that preserves vector addition and scalar multiplication. Example: $T(x) = Ax$ where $A$ is a matrix and $x$ is a vector.

• Kernel of a Transformation: The set of all vectors mapped to the zero vector by a linear transformation. Formula: $\ker(T) = \{v \in V : T(v) = 0\}$

• Matrix Diagonalization: The process of finding a diagonal matrix similar to a given square matrix, using its eigenvalues and eigenvectors. Formula: $A = PDP^{-1}$ where $D$ is diagonal and $P$ contains eigenvectors.

• Eigenvalues and Eigenvectors: For a square matrix $A$, an eigenvector $v$ and eigenvalue $\lambda$ satisfy $Av = \lambda v$.

• Determinant of a Matrix: A scalar value that can be computed from the elements of a square matrix and encodes certain properties such as invertibility. Formula: For $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\det(A) = ad - bc$.

• Cramer's Rule: A method for solving systems of linear equations using determinants. Formula: $x_i = \frac{\det(A_i)}{\det(A)}$ where $A_i$ is $A$ with the $i$th column replaced by the constants.

• Range of a Transformation: The set of all possible outputs (images) of a linear transformation.

• Matrix Rank: The dimension of the column space (or row space) of a matrix; indicates the number of linearly independent columns (or rows).

• Isomorphic Vector Spaces: Two vector spaces are isomorphic if there exists a bijective linear transformation between them.

• Inner Product Spaces: Vector spaces equipped with an additional structure called an inner product, allowing measurement of angles and lengths. Formula: For vectors $u, v$, $\langle u, v \rangle = u_1v_1 + u_2v_2 + \cdots + u_nv_n$.

• Gram-Schmidt Process: An algorithm for orthonormalizing a set of vectors in an inner product space.

• Basis of a Vector Space: A set of linearly independent vectors that span the entire space.

• Linear Independence: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.

• Equivalent Matrices: Matrices that can be transformed into each other by a sequence of elementary row operations.

• Row Echelon Form: A matrix form used to solve systems of linear equations; each leading entry is to the right of the leading entry in the previous row.

• Matrix Inverse: For a square matrix $A$, the inverse $A^{-1}$ satisfies $AA^{-1} = I$ where $I$ is the identity matrix.

• Solving Linear Systems: Methods include Gaussian elimination, Cramer's rule, and using matrix inverses.

Required Texts

Main Textbook

• Title: Linear Algebra

• Authors: Lay, David; McDonald, Judi; Lay, Steven

• Publisher: Pearson Publishing

• Edition: 6th Edition

• ISBN: 9780135851258

This textbook covers all major topics in linear algebra, including vector spaces, linear transformations, matrices, eigenvalues, and applications.

Grading Criteria and Major Assignments

Grade Calculation

Component	Value (%)
Attendance	5%
Homework (Pearson My Math Lab)	15%
Quizzes	5%
Project	20%
Exams (3)	30%
Final Exam	25%
Extra Credit (Tutoring)	Up to 5%

Grading Scale

Letter Grade	Percentage Range
A	90 - 100%
B	80 - 89%
C	70 - 79%
D	60 - 69%
F	0 - 59%

Course Policies and Procedures

Attendance and Participation

• Attendance is required and affects your grade.

• Excessive absences (excused or unexcused) may result in a lower grade or failure.

• Participation in class and discussion assignments is expected.

Technology Policy

• Laptops or desktops running Windows or macOS are required.

• Respondus Lockdown Browser is used for online quizzes and exams.

• Calculators (TI-84 recommended) may be used for homework and exams; know how to use your calculator.

• Cell phones and other devices with internet access are not allowed during exams.

Make-up Work and Late Policy

• No make-up tests or late assignments without approved university excuse.

• Documentation is required for excused absences due to illness or emergencies.

• Homework submitted late will be penalized: 10% if 1-3 days late, 20% if 4-7 days late, 30% if 4+ days late.

Summary Table: Key Linear Algebra Concepts

Concept	Definition	Example/Formula
Vector Space	Set of vectors with addition and scalar multiplication	$\mathbb{R}^n$
Linear Transformation	Function preserving vector operations	$T(x) = Ax$
Matrix	Rectangular array of numbers	$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$
Eigenvalue	Scalar $\lambda$ such that $Av = \lambda v$	$\lambda$
Eigenvector	Nonzero vector $v$ such that $Av = \lambda v$	$v$
Determinant	Scalar value encoding matrix properties	$\det(A)$
Rank	Dimension of column space	$\text{rank}(A)$
Basis	Linearly independent spanning set	$\{v_1, v_2, \ldots, v_n\}$
Row Echelon Form	Matrix form for solving equations	Gaussian elimination

Additional Info

• Linear Algebra is essential for advanced mathematics, engineering, computer science, and data science.

• Mastery of these concepts will prepare students for further study in mathematics and related fields.